Contents

### Context

#### 2-Category theory

2-category theory

# Contents

## Idea

Two morphisms $C \stackrel{L}{\to} D$ and $D \stackrel{R}{\to} C$ in a 2-category $\mathcal{C}$ form an adjunction if they are dual to each other (Lambek 82).

There are two archetypical examples:

• If $A$ is a monoidal category and $\mathcal{C} = \mathbf{B}A$ is the one-object 2-category incarnation of $A$ (the delooping of $A$), so that the morphisms in $\mathcal{C}$ correspond to the objects of $A$, then the notion of adjoint morphisms in $\mathcal{C}$ coincides precisely with the notion of dual objects in $A$.

• If $\mathcal{C}$ is the $2$-category Cat, so that the morphisms in $\mathcal{C}$ are functors, then the notion of adjoint morphisms in $\mathcal{C}$ coincides precisely with the notion of adjoint functors.

The notion of adjunction may usefully be thought of as a weakened version of the notion of equivalence in a 2-category: a morphism in an adjunction need not be invertible, but it has in some sense a left inverse from below and a right inverse from above. If the morphism in an adjunction does happen to be a genuine equivalence, then we speak of the adjunction being an adjoint equivalence.

Essentially everything that makes category theory nontrivial and interesting beyond groupoid theory can be derived from the concept of adjoint functors. In particular universal constructions such as limits and colimits are examples of certain adjunctions. Adjunctions are already interesting (but simpler) in 2-posets, such as the $2$-poset Pos of posets.

## Definition

### Direct definition

An adjunction in a 2-category is a pair of objects $C,D$ together with morphisms $L: C \to D$, $R : D \to C$ and 2-morphisms $\eta: 1_C \to R \circ L$, $\epsilon: L \circ R \to 1_D$ satisfying the equations

$(R \epsilon) \cdot (\eta R) = 1_R \qquad \text{i.e.} \qquad R \stackrel{\eta \circ 1_R}{\to} R \circ L \circ R \stackrel{1_R \circ \epsilon}{\to} R = R \stackrel{1_R}{\to} R \qquad \text{i.e.} \qquad \array{\arrayopts{ \padding{0} } &&&&1_C& \\ &&\cellopts{\colspan{5}}\begin{svg} [[!include adjunction > zigzageta]] \end{svg}\\ D & \stackrel{R}{\to}& C & \stackrel{L}{\to}& D & \stackrel{R}{\to}& C \\ \cellopts{\colspan{4}}\begin{svg} [[!include adjunction > zigzagepsilon]] \end{svg} \\ &&1_D& } \quad = \quad D \stackrel{R}{\to} C$

and

$(\epsilon L) \cdot (L \eta) = 1_L \qquad \text{i.e.} \qquad L \stackrel{1_L \circ \eta}{\to} L \circ R \circ L \stackrel{\epsilon \circ 1_L}{\to} L = L \stackrel{1_L}{\to} L \qquad \text{i.e.} \qquad \array{\arrayopts{ \padding{0} } &&1_C& \\ \cellopts{\colspan{5}}\begin{svg} [[!include adjunction > zigzageta]] \end{svg}\\ C & \stackrel{L}{\to}& D & \stackrel{R}{\to}& C & \stackrel{L}{\to}& D \\ &&\cellopts{\colspan{4}}\begin{svg} [[!include adjunction > zigzagepsilon]] \end{svg} \\ &&&&1_D& } \quad = \quad C \stackrel{L}{\to} D$

variously called the triangle identities or the zig-zag identities. We call $L$ the left adjoint (of $R$) and $R$ the right adjoint (of $L$). We call $\eta$ the unit of the adjunction and $\epsilon$ the counit of the adjunction.

When interpreted in the prototypical 2-category Cat, $C$ and $D$ are categories, $L$ and $R$ are functors, and $\eta$ and $\epsilon$ are natural transformations. In this case (which was of course the first to be defined) there are a number of equivalent definitions of an adjunction, which can be found on the page adjoint functor. Conversely, the definition in any 2-category can be obtained by internalization from the definition in $\Cat$.

### In terms of string diagrams

The definition of an adjunction may be nicely expressed using string diagrams. The data $L: C \to D$, $R : D \to C$ and 2-cells $\eta: 1_C \to R \circ L$, $\epsilon: L \circ R \to 1_D$ are depicted as

(where 1-cells read from right to left and 2-cells from bottom to top), and the zigzag identities are expressed as “pulling zigzags straight” (hence the name):

Often, arrows on strings are used to distinguish $L$ and $R$, and most or all other labels are left implicit; so the zigzag identities, for instance, become:

## Examples

At the cost of some repetition (compare adjoint functor), we outline how one gets from the hom-functor formulation of adjunction in Cat to the elementary definition in terms of units and counits. This will motivate the definition in the section that follows, which is elementary (definable in the first-order theory of categories) and portable to any 2-category.

We start from a familiar example. Let $U: Grp \to Set$ from groups to sets be the usual forgetful functor. When we say “$F(X)$ is the free group generated by a set $X$”, we mean there is a function $\eta_X: X \to U(F(X))$ which is universal among functions from $X$ to the underlying set of a group, which means in turn that given a function $f: X \to U(G)$, there is a unique group homomorphism $g: F(X) \to G$ such that

$f = (X \stackrel{\eta_X}{\to} U(F(X)) \stackrel{U(g)}{\to} U(G))$

Here $\eta_X$ is a component of what we call the unit of the adjunction $F \dashv U$, and the equation above is a recipe for the relationship between the map $g: F(X) \to G$ and the map $f: X \to U(G)$ in terms of the unit.

Now we work more generally. Suppose given functors $L: C \to D$, $R: D \to C$ and the structure of an adjunction in the form of a natural isomorphism

$\Psi_{c, d}: \hom_D(L(c), d) \cong \hom_C(c, R(d))$

Now the idea is that, in the spirit of the (proof of the) Yoneda lemma, we would like $\Psi$ to be determined by what it does to identity maps. With that in mind, define the unit $\eta : 1_C \to R L$ by the formula $\eta_c = \Psi_{c, L(c)}(1_{L(c)})$. Dually, define the counit $\varepsilon : L R \to 1_D$ by the formula $\varepsilon_d = \Psi^{-1}_{R(d), d}(1_{R(d)})$. Then given $g: L(c) \to d$, the claim is that

$\Psi_{c, d}(g) = (c \stackrel{\eta_c}{\to} R(L(c)) \stackrel{R(g)}{\to} R(d)).$

This may be left as an exercise in the yoga of the Yoneda lemma, applied to $\hom_D(L(c), -) \to \hom_C(c, R(-))$. By duality, given $f: c \to R(d)$,

$\Psi^{-1}_{c, d}(f) = (L(c) \stackrel{L(f)}{\to} L(R(d)) \stackrel{\varepsilon_d}{\to} d).$

(In fact, we spell out the Yoneda-lemma proof of this dual form below.)

Finally, these operations should obviously be mutually inverse, but that can again be entirely encapsulated Yoneda-wise in terms of the effect on identity maps. Thus, if $\eta_c \coloneqq \Psi_{c, L(c)}(1_{L(c)})$, via the recipe just given for $\Psi^{-1}$ we recover

$1_{L(c)} = (L(c) \stackrel{L(\eta_c)}{\to} L R L(c) \stackrel{\varepsilon_{L(c)}}{\to} L(c))$

and this is one of the famous triangle identities: $1_L = (L \stackrel{L \eta}{\to} L R L \stackrel{\varepsilon L}{\to} L)$. Here, juxtaposition of functors and natural transformations denotes neither functor application, nor vertical composition, nor horizontal composition, but whiskering. By duality, we have the other triangle identity $1_R = (R \stackrel{\eta R}{\to} R L R \stackrel{R \varepsilon}{\to} R)$. These two triangular equations are enough to guarantee that the recipes for $\Psi$ and $\Psi^{-1}$ indeed yield mutual inverses.

Thus, it is perfectly sufficient to define an adjoint pair of functors in $Cat$ as given by unit and counit transformations $\eta: 1_C \to R L$, $\varepsilon: L R \to 1_D$, satisfying the triangle identities above.

The definition of adjunctions via units and counits is an “elementary” definition (so that by implication, the formulation in terms of hom-functors is not elementary) in the sense that while the hom-functor formulation relies some notion of hom-_set_, the formulation in terms of units and counits is purely in the first-order language of categories and makes no reference to a model of set theory. (Cf. first-order logic and second-order logic.) The definition via (co)units therefore gives us a definition of adjunctions even if an assumption of local smallness is not made.

###### Yoneda-lemma argument

We claim that $\Psi^{-1}_{c, d}: \hom_C(c, R(d)) \to \hom_D(L(c), d)$ can be defined by the formula

$\Psi^{-1}_{c, d}(f: c \to R(d)) = (L(c) \stackrel{L(f)}{\to} L(R(d)) \stackrel{\varepsilon_d}{\to} d)$

where $\varepsilon_d \coloneqq \Psi^{-1}_{R(d), d}(1_{R(d)})$. This is by appeal to the proof of the Yoneda lemma applied to the transformation

$\Psi^{-1}_{-, d}: \hom_C(-, R(d)) \to \hom_D(L(-), d)$

For the naturality of $\Psi^{-1}$ in the argument $(-)$ would imply that given $f: c \to R(d)$, we have a commutative square

$\array{ \hom_C(R(d), R(d)) & \stackrel{\Psi^{-1}_{R(d), d}}{\to} & \hom_D(L(R(d)), d) \\ \hom_C(f, R(d)) \downarrow & & \downarrow \hom_D(L(f), d) \\ \hom_C(c, R(d)) & \underset{\Psi^{-1}_{c, d}}{\to} & \hom_D(L(c), d) }$

Chasing the element $1_{R(d)}$ down and then across, we get $f: c \to R(d)$ and then $\Psi^{-1}_{c, d}(f)$. Chasing across and then down, we get $\varepsilon_d$ and then $\varepsilon_d \circ L(f)$. This completes the verification of the claim.

### More examples

• For $A$ a monoidal category and $\mathbf{B}A$ its incarnation as a one-object 2-category (its delooping), an adjunction in $\mathbf{B}A$ is a pair of dual objects.

• For $\mathcal{C}$ a 2-groupoid, an adjunction in $\mathcal{C}$ is an adjoint equivalence.

## References

For the basics, see any text on category theory (and see the references at adjoint functor), for instance:

For some early history and illustrative examples see

• Joachim Lambek, The Influence of Heraclitus on Modern Mathematics, In Scientific Philosophy Today: Essays in Honor of Mario Bunge, edited by Joseph Agassi and Robert S Cohen, 111–21. Boston: D. Reidel Publishing Co. (1982) (doi:10.1007/978-94-009-8462-2_6)

(more along these lines at objective logic).

The fundamental role of adjunctions in logic/type theory originates with the observaiton that substitution forms an adjoint triple with existential quantification and universal quantification: